Chapter 3
Hyperbolicity
3.1 The stable manifold theorem
In this section we shall mix two typical ideas of hyperbolic dynamical systems: the use of a linear approximation to understand the behavior of nonlinear systems, and the examination of the behavior of the system
along a reference orbit.
The latter idea is embodied in the notion of stable set.
Definition 3.1.1: Let (X, f ) be a dynamical system on a metric space X. The stable set of a point x ∈ X is
½
¾
¯
¡
¢
¯
Wfs (x) = y ∈ X ¯ lim d f k (y), f k (x) = 0 .
k→∞
In other words, y ∈
if and only if the orbit of y is asymptotic to the orbit of x. If no confusion will
arise, we drop the index f and write just W s (x) for the stable set of x.
Wfs (x)
Clearly, two
or are disjoint; therefore X is the disjoint union of its stable sets.
¡ coincide
¢
¡ stable¢sets either
Furthermore, f W s (x) ⊆ W s f (x) for any x ∈ X.
The twin notion of unstable set for homeomorphisms is defined in a similar way:
Definition 3.1.2: Let f : X → X be a homeomorphism of a metric space X. Then the unstable set of a
point x ∈ X is
½
¾
¯
¡
¢
¯
Wfu (x) = y ∈ X ¯ lim d f −k (y), f −k (x) = 0 .
k→∞
In other words, Wfu (x) = Wfs−1 (x).
Again, unstable sets of homeomorphisms give a partition of the space. On the other hand, stable sets
and unstable sets (even of the same point) might intersect, giving rise to interesting dynamical phenomena.
Remark 3.1.1. If f is not invertible, the definition of the unstable set is more complicated. Let (X, f ) be
a dynamical system. The dynamical completion of (X, f ) is the dynamical system (X̂, fˆ), where
ª
©
X̂ = x̂ = {xk } ∈ X Z | xk+1 = f (xk ) for all k ∈ Z ,
and fˆ is just the left shift fˆ({xk }) = {f (xk )}. Since f is continuous, X̂ is a closed subset of X Z , and fˆ is
continuous with respect to the induced topology. More precisely, fˆ is a homeomorphism of X̂. Furthermore,
the completion comes equipped with a canonical projection π: X̂ → X given by π(x̂) = x0 , so that f ◦π = π◦fˆ.
In a very precise sense (see Exercise 3.1.1), the completion is the least possible invertible extension of (X, f ).
We also explicitely remark that if X is a metric space then X̂ inherites a distance function too. Given x ∈ X,
an history of x is a x̂ ∈ X̂ such that π(x̂) = x. Notice that if x̂ ∈ X̂ then its positive half {xk }k∈N simply is
the orbit of x0 ; on the other hand, its negative half is just a possible backward orbit of x0 . In particular, π
is a homeomorphism if and only if f is a homeomorphism (and it is onto if and only if f is, and injective if
and only if f is). Finally, for every history x̂ ∈ X̂ of x ∈ X the unstable set of x with respect to x̂ is
¡
¢
Wfu (x; x̂) = π Wfuˆ (x̂) .
In other words, Wfu (x; x̂) is the set of y ∈ X such that there exists a backward orbit of y which is asymptotic
to the chosen backward orbit of x. So if f is not invertible there are in general as many unstable sets as
there are backward orbits (and they do not form a partition of X).
44
Sistemi Dinamici Discreti, A.A. 2005/06
Exercise 3.1.1. State the universal property enjoyed by the triple (X̂, fˆ, π), and use it to prove that the
dynamical completion is unique up to topological conjugation.
For the sake of simplicity, in this section we shall limit ourselves to invertible maps only; however, using
the dynamical completion it is possible to extend almost all the results we shall prove to non-invertible maps.
Exercise 3.1.2.
If x0 is a periodic point of period n of a homeomorphism f : X → X prove that
n−1
[
k=0
[
¢ n−1
¢
¡
¡
Wfs f k (x0 ) =
Wfsn f k (x0 ) and
k=0
n−1
[
Wfu (x0 ) =
k=0
n−1
[
¢
¡
Wfun f k (x0 ) .
k=0
If L is a hyperbolic linear map, then the stable and unstable sets of the origin are exactly the stable and
unstable subspaces E s/u (L); see Theorem 1.3.5. In particular, they are smooth and intersect transversally.
The main goal of this section is to prove that this is a general feature: if x0 is a hyperbolic fixed point (see
below for the definition) of a smooth dynamical system (M, f ), where M is a smooth manifold, then the
stable and unstable sets are actually manifolds as smooth as the map f , and intersecting transversally in x0 .
Definition 3.1.3: Let (M, f ) be a C 1 dynamical system on a smooth manifold M . Then a periodic point p
of period n is hyperbolic if the linear map d(f n )p : Tp M → Tp M is a hyperbolic linear map. Its orbit will be
called a hyperbolic periodic orbit.
Definition 3.1.4: Let (X, f ) be a dynamical system on a metric space X. Let p be a fixed point of f ,
and B(p, δ) the open ball of radius δ > 0 centered in p. Then the local stable set for f at p of radius δ is
Wfs (p, δ) = {x ∈ B(p, δ) | x ∈ Wfs (p) and f k (x) ∈ B(p, δ) for all k ∈ N}.
Similarly, if f is a homeomorphism the local unstable set for f at p of radius δ is
Wfu (p, δ) = {y ∈ B(p, δ) | y ∈ Wfu (p) and f −k (y) ∈ B(p, δ) for all k ∈ N}.
Definition 3.1.5: Let L: V → V be a hyperbolic linear self-map of a vector space V on the field K = R or C.
The contraction rate λ(L) and the expansion rate µ(L) of L are defined by
λ(L) = r(L|E s ) = sup{|χ| | χ ∈ sp(L), |χ| < 1},
¢−1
¡
= inf{|χ| | χ ∈ sp(L), |χ| > 1}.
µ(L) = r (L|E u )−1
We are now ready to state the main results of this section.
Theorem 3.1.1: (Stable manifold theorem) Let f : M → M be a C r diffeomorphism, with r ≥ 1, of a
Riemannian manifold M . Let p ∈ M be a hyperbolic fixed point of f . Then there is δ > 0 such that:
(i) the
stable
set W s (p, δ) is an embedded C r submanifold such that Tp W s (p, δ) = E s (dfp );
¡ local
¢
s
(ii) f W (p, δ) ⊆ W s (p, δ);
(iii) for any ε > 0 there is Cε > 0 such that
¡
¢
£
d f k (y), p ≤ Cε λ(dfp ) + ε]k d(y, p)
for all y ∈ W s (p, δ) and k ∈ N, where d is the distance induced by the Riemannian metric;
∈ B(p, δ)¡for all k ∈
(iv) y ∈ W s (p, δ) if and only if f k (y)S
¢ N;
(v) the global stable set is given by k∈N f −k W s (p, δ) , and thus it is a C r immersed submanifold.
Remark 3.1.2. If f is not invertible, the same statement holds except for the last assertion:
the global¢
¡
stable set might not be a submanifold. For instance, if f : R2 → R2 is given by f (x, y) = 2x(y + 1), x(y + 1)
then the stable set of the origin is {x = 0} ∪ {y = −1}.
Clearly, an analogous statement holds for the unstable sets.
Remark 3.1.3. Because of the previous theorem, the stable/unstable set of a hyperbolic fixed point is
called the stable/unstable manifold. Notice that Exercise 3.1.2 implies that similar results hold for stable
and unstable sets of hyperbolic periodic points.
We shall describe a proof of Theorem 3.1.1 due to M.C. Irwin, and working in the setting of Banach
spaces.
3.1
The stable manifold theorem
45
Definition 3.1.6: Let E = E1 ⊕E2 be a splitting of a Banach space E (where here and in the rest of this section
E1 and E2 are always closed subspaces); for j = 1, 2 we shall denote by pj : E → Ej the linear projection
of E onto Ej , and we shall endow E with the box norm kxk = max{kp1 (x)k, kp2 (x)k}. Furthermore, if r > 0
we denote by Ej (r) the closed ball of radius r centered at the origin in Ej ; in particular, the closed ball of
radius r centered at the origin in E (endowed with the box norm) can be identified with E1 (r) × E2 (r).
Definition 3.1.7: Let T : E → E be a linear automorphism of E = E1 ⊕ E2 preserving the splitting, that is
such that T (E1 ) = E1 and T (E2 ) = E2 , and take λ ∈ (0, 1). We say that T is λ-hyperbolic (with respect to
the given splitting) if
and
kT2−1 k < λ,
kT1 k < λ
where Tj = T |Ej for j = 1, 2.
Lemma 3.1.2: Let E = E1 ⊕ E2 be a splitting of a Banach space E. Given r > 0, let f : E1 (r) × E2 (r) → E
be a Lipschitz map such that Lip(f − T ) ≤ ε < 1 − λ for a suitable λ-hyperbolic linear automorphism T .
Then, setting fj := pi ◦ f , if x = (x1 , x2 ), y = (y1 , y2 ) ∈ E1 (r) × E2 (r) are such that kx1 − y1 k ≤ kx2 − y2 k,
then
kf1 (x) − f1 (y)k ≤ (λ + ε)kx2 − y2 k < (λ−1 − ε)kx2 − y2 k ≤ kf2 (x) − f2 (y)k.
Proof : First of all we have
kf1 (x) − f1 (y)k ≤ kp1 (f − T )(x) − p1 (f − T )(y)k + kT1 (x1 ) − T1 (y1 )k
≤ εkx − yk + λkx1 − y1 k ≤ (λ + ε)kx − yk
≤ (λ + ε)kx2 − y2 k,
(3.1.1)
because we are using the box norm.
On the other hand,
kf2 (x) − f2 (y)k ≥ kT2 (x2 ) − T2 (y2 )k − kp2 (f − T )(x) − p2 (f − T )(y)k
≥ λ−1 kx2 − y2 k − εkx − yk
= (λ−1 − ε)kx2 − y2 k.
To conclude, it suffices to observe that λ + ε < 1 < λ−1 − ε.
Remark 3.1.4.
Even without assuming that kx1 − y1 k < kx2 − y2 k, the computations in (3.1.1) yields
kf1 (x) − f1 (y)k ≤ (λ + ε)kx − yk.
(3.1.2)
Definition 3.1.8: Given r > 0 and a map f : E1 (r) × E2 (r) → E, we define the local stable set of f by
∞
\
©
ª
¢
¡
f −k E1 (r) × E2 (r) .
W s (f, r) = x ∈ E1 (r) × E2 (r) | kf k (x)k ≤ r for all k ≥ 0 =
k=0
Clearly W s (f, r) is f -invariant and contains all fixed points of f in E1 (r) × E2 (r); however, it might be
empty.
Remark 3.1.5. We shall see that if the origin is a hyperbolic fixed point for f and r is small enough,
then the set W s (f, r) coincides with the local stable set of f at the origin of radius r introduced in the
Definition 3.1.4.
Proposition 3.1.3: Let E = E1 ⊕ E2 be a splitting of a Banach space E. Given 0 < r < +∞,
let f : E1 (r) × E2 (r) → E be a Lipschitz map such that Lip(f − T ) ≤ ε < 1 − λ for a suitable λ-hyperbolic
linear automorphism
T .¢ Then the set W s (f, r) is the graph of a Lipschitz map g: A → E2 (r) with Lip(g) ≤ 1,
¡
where A = p1 W s (f, r) ⊆ E1 (r). Furthermore, f |W s (f,r) is a contraction. In particular, f has at most one
fixed point which, if it exists, attracts exponentially all other points of W s (f, r).
Proof : To prove the existence of g it suffices to show that for any pair x = (x1 , x2 ) and y = (y1 , y2 ) of points
of W s (f, r) we have kx2 − y2 k ≤ kx1 − y1 k. Indeed, if this happens then p−1
1 (x1 ) must consist of exactly one
point x2 = g(x1 ) for all x1 ∈ A, and g: A → E2 (r) is Lipschitz of constant at most 1.
46
Sistemi Dinamici Discreti, A.A. 2005/06
Let us assume, by contradiction, that we had kx2 − y2 k > kx1 − y1 k; using the previous lemma we get
kf2 (x) − f2 (y)k ≥ (λ−1 − ε)kx2 − y2 k > (λ + ε)kx2 − y2 k ≥ kf1 (x) − f1 (y)k,
and then, by induction,
kp2 ◦ f k (x) − p2 ◦ f k (y)k ≥ (λ−1 − ε)k kx2 − y2 k
for all k ∈ N. But, by definition of W s (f, r), we have kp2 ◦ f k (x) − p2 ◦ f k (y)k ≤ 2r for all k ∈ N;
since (λ−1 − ε)k → +∞ as k → ∞, we would have to conclude that kx2 − y2 k = 0, against our assumption.
We are left to proving that f |W s (f,r) is a contraction. We have seen that x, y ∈ W s (f, r) implies kx−yk = kx1 −y1 k. Moreover, since W s (f, r) is f -invariant, we also have kf (x)−f (y)k = kf1 (x)−f1 (y)k.
But then (3.1.2) yields
kf (x) − f (y)k = kf1 (x) − f1 (y)k ≤ (λ + ε)kx − yk;
so f is a contraction on W s (f, r), and the rest is a trivial consequence of Theorem 1.2.1.
Of course, this is not enough to get a stable manifold theorem: we should at least prove that A is not
empty. To do so we need a few ausiliary results.
Lemma 3.1.4: Let f , g: X → F be continuous maps from a metric space X to a Banach space F . Let us
suppose that f is injective with inverse f −1 Lipschitz, and that Lip(g − f ) Lip(f −1 ) < 1. Then g is injective,
g −1 is Lipschitz and
n£
o−1
¤−1
Lip(f −1 )
.
− Lip(f − g)
=
Lip(g −1 ) ≤ Lip(f −1 )
1 − Lip(g − f ) Lip(f −1 )
Proof : To prove the assertion it is enough to observe that:
kg(x) − g(y)k ≥ kf (x) − f (y)k − k(f − g)(x) − (f − g)(y)k
o
n£
¢¤−1
¡
− Lip(g − f ) d(x, y).
≥ Lip f −1
Theorem 3.1.5: (Lipschitz inverse function theorem) Let E and F be two Banach spaces, U an open set
in E, V an open set in F , and f : U → V a homeomorphism with a Lipschitz inverse. Let, moreover, h: U → F
be a Lipschitz map such that Lip(h) Lip(f −1 ) < 1. Then g = h + f is a homeomorphism, with a Lipschitz
inverse, from U onto an open set of F . Moreover, if, in addition, E = F , the map f is a linear automorphism
of the Banach space E, and h is of class C r (with r ≥ 1), then g −1 is of class C r too.
Proof : By the previous lemma we already know that g is injective with Lipschitz inverse; we need to prove
that g is open, which is equivalent to saying that g ◦ f −1 = id +h ◦ f −1 is open, since f is a homeomorphism.
Notice that if v = h ◦ f −1 and λ = Lip(v) we have
λ ≤ Lip(h) Lip(f −1 ) < 1.
Now let x be in V , and r > 0 such that Br (x) ⊆ V , where Br (x) is the closed ball of center x and radius r.
We claim that
¢
¡
¢
¡
(3.1.3)
(id +v) Br (x) ⊇ B(1−λ)r (id +v)(x) .
To prove this we can clearly assume x = v(x) = O. It suffices then to find a map w: Bs (O) → F such that
¢
¡
and
(id +v) ◦ (id +w) = id,
(id +w) Bs (O) ⊆ Br (O)
where s = (1 − λ)r. Notice that we must have w = −v ◦ (id + w); so we try to get w as fixed point of a
suitable operator.
Let
¾
½
¢¯
¡
λ
0
¯
,
Z := w ∈ C Bs (O), F w(O) = O and Lip(w) ≤
1−λ
¡
¢
which is complete with the sup-norm. Moreover it is easy to see that if w ∈ Z then (id +w) Bs (O) ⊆ Br (O)
and −v ◦ (id +w) ∈ Z. So the operator Φ(w) := −v ◦ (id +w) sends Z into itself; furthermore
kΦ(w) − Φ(w0 )k = k − v ◦ (id +w) + v ◦ (id +w0 )k ≤ λkw − w0 k,
and thus Φ is a contraction of Z. So it has a unique fixed point in Z, (3.1.3) is proved, and g is open.
The last assertion is well known.
3.1
The stable manifold theorem
47
Corollary 3.1.6: Let U ⊆ E be an open subset of a Banach space E, and g a homeomorphism from U onto
an open set of a Banach space F . If g −1 is Lipschitz and λ ≥ Lip(g −1 ) then
¡
¢
¡
¢
g Br (x) ⊇ Br/λ g(x)
for all x ∈ U and r > 0 such that Br (x) ⊆ U .
Proof : We can again suppose x = g(x) = 0. Let v 6= O be a point of the closed ball Br/λ (O), and define
¯
©
¡
¢ª
t∞ := sup t ≥ 0 ¯ [0, t]v ⊂ g Br (O) .
¡
¢
¡
¢
Since g Br (O) contains a neighbourhood of the origin, we have that t∞ > 0; moreover [0, t∞ )v ⊂ g Br (O) .
¡
¢
The continuity of g −1 then ensures that lim g −1 (tv) ∈ Br (O); and thus t∞ v ∈ g Br (O) .
t→t∞
Now, to prove the assertion we have only to show that t∞ ≥ 1. To do this let us suppose t∞ < 1; then
kg −1 (t∞ v)k ≤ λ t∞ kvk < r,
¡
¢
and ¡hence ¢t∞ v ∈ g Br (O) . Thus it is possible to find ε > 0 such that [t∞ , t∞ + ε)v is still contained
in g Br (O) , which contradicts the maximality of t∞ .
We are finally ready to prove the
Theorem 3.1.7: (Stable Manifold Theorem for Banach spaces) Let E = E1 ⊕ E2 be a splitting of a Banach
space E, and T : E → E be a λ-hyperbolic linear automorphism of E. Then there is an ε > 0, depending
only on λ, such that for all r > 0 there exists a δ > 0 so that if f : E1 (r) × E2 (r) → E satisfies Lip(f − T ) < ε
and kf (O)k < δ, then W s (f, r) is the graph of a Lipschitz function g: E1 (r) → E2 (r) with Lip(g) ≤ 1.
Moreover, if f is of class C k (with k ≥ 1) then so is g.
Proof : Observe preliminarily that for x = (x1 , x2 ) ∈ W s (f, r), the sequence γk = f k+1 (x) satisfies the
following conditions:
(i) kγk k ≤ r for all k ≥ 0;
(ii) f (γk ) − γk+1 = O for all k ≥ 0;
(iii) f (x) − γ0 = O.
Conversely, if x ∈ E(r) and {γk }k≥0 satisfy the previous conditions, then x ∈ W s (f, r) and γk = f k+1 (x).
Now we develop this idea. Let us consider the Banach space
½
¾
¯
∞
N ¯
sup kγk k < ∞ ,
` (E) = γ ∈ E
k∈N
endowed with the norm
kγk`∞ (E) = sup kγk k.
k∈N
Let B(r) be the closed ball of radius r and center the origin in `∞ (E). Endow E × `∞ (E) and E1 × `∞ (E)
with the box norms induced by E, E1 , and `∞ (E).
Now let F: E1 (r) × E2 (r) × B(r) → E1 × `∞ (E) be the map defined by
¡
¢
F(x1 , x2 , γ) = x1 , Fx1 (x2 , γ) ,
where Fx1 : E2 (r) × B(r) → `∞ (E) is given by
¡
¢
Fx1 (x2 , γ)
k
½
=
f (x1 , x2 ) − γ0 , if k = 0,
if k ≥ 1,
f (γk−1 ) − γk
so that Fx1 (x2 , γ) = O if and only if γk = f k+1 (x1 , x2 ) for all k ∈ N. If we show that F is invertible and that
its image contains E1 (r) × {O}, it would follow that for every x1 ∈ E1 (r) there exists a unique x2 ∈ E2 (r)
48
Sistemi Dinamici Discreti, A.A. 2005/06
such that (x1 , x2 ) ∈ W s (f, r), and hence we can define g as π2 ◦ F −1 |E1 (r)×{O} , where π2 is the projection
of E1 × E2 × `∞ (E) onto E2 . We shall of course use the Lipschitz inverse function theorem to do this.
Define T : E1 × E2 × `∞ (E) → E1 × `∞ (E) as we did for F, replacing f by T , i.e., T (x1 , x2 , γ) = (x1 , ν)
where
½
T (x1 , x2 ) − γ0 if k = 0,
νk = Tx1 (x2 , γ)k =
T (γk−1 ) − γk for k ≥ 1.
It is clear that T is a linear operator, and it is easy to see that kT k ≤ 1 + kT k, so T is also continuous.
Moreover Lip(F − T ) ≤ Lip(f − T ). Consequently if T is invertible and Lip(f − T ) ≤ kT −1 k−1 , using the
Lipschitz inverse function theorem we obtain the invertibility of F.
Let us prove that T is invertible and compute its inverse. We need to express x2 and γ in terms of x1
and ν. Writing γk = (γk,1 , γk,2 ) ∈ E1 × E2 and νk = (νk,1 , νk,2 ) ∈ E1 × E2 , the definition of ν becomes:

ν0,1 = T1 (x1 ) − γ0,1 ,


ν0,2 = T2 (x2 ) − γ0,2 ,

 νk,1 = T1 (γk−1,1 ) − γk,1
νk,2 = T2 (γk−1,2 ) − γk,2
for k ≥ 1,
for k ≥ 1.
From the first and the third equations we obtain an expression for γk,1 :
γk,1 = T1k+1 (x1 ) −
k
X
T1k−j (νj,1 ).
j=0
The fourth equation for k ≥ 0 gives:
¢
¡
γk,2 = T2−1 (νk+1,2 + γk+1,2 ) = T2−1 νk+1,2 + T2−1 (νk+2,2 + γk+2,2 ) = · · ·
so in the limit
γk,2 =
∞
X
T2−j (νk+j,2 ).
j=1
Finally from the second equation
x2 =
∞
X
−(j+1)
T2
(νj,2 ).
j=0
Obviously these series converge; thus we have proved that T is invertible, and it can be easily checked
that kT −1 k ≤ (1 − λ)−1 .
Choose ε ≤ 1 − λ. The Lipschitz inverse function theorem then implies that if Lip(f − T ) < ε then F
is invertible, because
Lip(F − T ) ≤ Lip(f − T ) < ε ≤ kT −1 k−1 .
Now we prove that the image of F contains E1 (r) × {O} when ε and f (O) are small enough. Now,
the image of F contains (x1 , O) if and only if the image of Fx1 contains O. Furthermore Fx1 is a Lipschitz
perturbation of Tx1 and Lip(Fx1 − Tx1 ) ≤ Lip(f − T ). Since Tx1 differs from TO only by an additive constant,
we have that Lip(Fx1 − TO ) ≤ Lip(f − T ).
The above computations show that TO is invertible, with inverse TO−1 (ν) = (x2 , γ) given by
x2 =
∞
X
−(j+1)
T2
(νj,2 ),
j=0
γk,1 = −
γk,2 =
k
X
T1k−j (νj,1 ),
j=0
∞
X −j
T2 (νk+j,2 ),
j=1
3.1
The stable manifold theorem
49
so that kTO−1 k ≤ (1 − λ)−1 .
If Lip(f − T ) < ε ≤ 1 − λ, we get Lip(TO−1 Fx1 − id) ≤ ε/(1 − λ) < 1 and using Lemma 3.1.4 we obtain
Lip
h¡
TO−1 Fx1
¢−1 i
1
ε ;
1 − 1−λ
≤
then Corollary 3.1.6 yields
TO−1 Fx1 [E2 (r) × B(r)] ⊇ TO−1 Fx1 (O, O) + E2 (s) × B(s),
where s = r(1 − ε/(1 − λ)).
Now we compute kTO−1 Fx1 (O, O)k. First of all we have Fx1 (O, O) = ν ∈ `∞ (E), with ν0 = f (x1 , O)
and νk = f (O, O) for k ≥ 1. Consequently, TO−1 Fx1 (O, O) = (x2 , γ), where
∞
¢ X
¢
¡
¡
T2−j f2 (O, O) ,
x2 = T2−1 f2 (x1 , O) +
j=2
k
¢ X
¢
¡
¡
γk,1 = −T1k f1 (x1 , O) −
T1k−j f1 (O, O) ,
j=1
γk,2 =
∞
X
¢
¡
T2−j f2 (O, O) .
j=1
Now we have
and further
kf1 (x1 , O)k ≤ kp1 (f − T )(x1 , O)k + kT1 (x1 )k
≤ k(f − T )(x1 , O)k + λkx1 k,
kf2 (x1 , O)k ≤ kp2 (f − T )(x1 , O)k ≤ k(f − T )(x1 , O)k,
k(f − T )(x1 , O)k ≤ k(f − T )(O)k + k(f − T )(x1 , O) − (f − T )(O)k
≤ kf (O)k + εkx1 k.
Putting all together we get
kTO−1 Fx1 (O)k ≤ (λ + ε)r +
1
kf (O)k.
1−λ
Therefore the image of TO−1 Fx1 contains the ball centered at TO−1 Fx1 (O) of radius s = r(1 − ε/(1 − λ));
1
kf (O)k < s. We can rewrite this inequality as
hence it will contain O whenever (λ + ε)r + 1−λ
µ
kf (O)k < (1 − λ)r 1 − λ − ε −
ε
1−λ
¶
= r[(1 − λ)2 − ε(2 − λ)].
Suppose then ε < (1 − λ)2 /(2 − λ) < 1 − λ and take δ > 0 so that δ < r[(1 − λ)2 − ε(2 − λ)]; if Lip(f − T ) < ε
and kf (O)k < δ, we have shown that TO−1 Fx1 contains O. Since TO is linear, the image of Fx1 contains O
as well, and so the image of F contains E1 (r) × {O}, as desired.
In conclusion we have proved the following:
If Lip(f − T ) < ε < (1 − λ)2 /(2 − λ) and kf (O)k < δ < r[(1 − λ)2 − ε(2 − λ)], then W s (f, r) is the
graph of a Lipschitz map g = π2 ◦ F −1 |E1 (r)×{O} : E1 (r) → E2 (r) with Lip(g) ≤ 1.
To finish the proof of the stable manifold theorem we only need to show that g is C k whenever f is C k .
By definition we have that g is C k if F −1 is so. We saw that Lip(F − T ) < kT −1 k−1 ; therefore, by the
Lipschitz inverse function theorem, F −1 is C k whenever F is.
We are reduced then to showing that F is C k when f is C k . Unfortunately, `∞ (E) is too large for this
to be possible, but we can easily bypass this difficulty.
50
Sistemi Dinamici Discreti, A.A. 2005/06
The obvious candidate for the derivative of F at the point
(x, γ) = (x1 , x2 , γ) ∈ E1 (r) × E2 (r) × B(r)
is the linear map L: E1 × E2 × `∞ (E) → E1 × `∞ (E) given by L(y1 , y2 , ν) = (y1 , ζ), where
½
ζk =
dfx (y) − ν0
dfγk−1 (νk−1 ) − νk
for k = 0,
for k ≥ 1.
We have F(x + y, γ + ν) − F(x, γ) − L(y, ν) = (O, ϕ), where

R1

 ϕ0 = f (x + y) − f (x) − dfx (y) = 0 (dfx+ty − dfx ) (y) dt,
ϕk = f (γk−1 + νk−1 ) − f (γk−1 ) − dfγk−1 (νk−1 )
¢
R1¡


= 0 dfγk−1 +tνk−1 − dfγk−1 (νk−1 ) dt,
This shows that the quotient
for k ≥ 1.
kF(x + y, γ + ν) − F(x, γ) − L(y, ν)k
k(y, ν)k
is bounded from above by
µZ
Z
1
1
kdfx+ty − dfx k dt, sup
max
k−1 +tνk−1
k≥1
0
°
°dfγ
¶
°
°
− dfγk−1 dt .
0
It is clear that the supremum of all these integrals does not necessarily tend to zero with k(y, ν)k, since `∞ (E)
is not locally compact and df is not necessarily uniformly continuous. If we knew that {γk } were contained
in a compact subset of E(r), for instance if γk were convergent, then the supremum would go to zero
with k(y, ν)k. This leads us to consider the subspace C of convergent sequences:
½
C=
¾
∞
γ ∈ ` (E) | lim γk exists .
k→∞
∞
Since E is complete, C is closed
¡ in ` (E), and thus ¢C itself is a Banach space.
Notice that, obviously, F E1 (r) × E2 (r) × C(r) ⊆ E1 (r) × C; so it make sense to consider the restriction F̃ of F to E1 (r) × E2 (r) × C(r) and to define T̃ in the same way as the mapping induced by T . The
previous argument then shows that F̃ is C 1 , and by induction it can be proved that F̃ is C r when f is C r .
Now, we can repeat our previous construction of g with F̃, T̃ and C in place of F, T and `∞ (E),
provided that we know T̃ is invertible, i.e., that T −1 (E1 × C) ⊂ E1 × E2 × C. We proceed to check this.
Let (x, ν) be in E1 × C and recall that T −1 (x, ν) = (x1 , x2 , γ) is given by
x2 =
∞
X
−(j+1)
T2
(νj,2 ),
j=0
γk,1 = T1k+1 (x1 ) −
k
X
T1k−j (νj,1 ),
j=0
γk,2 =
∞
X
T2−j (νk+j,2 ).
j=1
To show that γ belongs to C we shall prove that γ1 and γ2 are Cauchy. Let us begin with γ1 :
°
°
° h
°
k
X
°X h−j
°
k−j
h+1
k+1
°
T1 (νj,1 ) −
T1 (νj,1 )°
kγh,1 − γk,1 k ≤ °
° + kT1 (x1 ) − T1 (x1 )k.
° j=0
°
j=0
3.1
The stable manifold theorem
51
Since kT1 k < λ < 1, the term kT1h (x1 ) − T1k (x1 )k goes to zero as h and k go to ∞.
For the other term, setting N = min{h, k}, we have
°
° °
°
° h
° ° h
°
k
k
X
X
X
°X h−j
°
°
°
k−j
j
j
°
°
°
T1 (νj,1 ) −
T1 (νj,1 )° = °
T1 (νh−j,1 ) −
T1 (νk−j,1 )°
°
°
° j=0
° °j=0
°
j=0
j=0


à ∞
!
∞
X
X
j
k

λ
λ kν1 k.
sup kνr,1 − νs,1 k +
≤
r≥h−N
j=0
k=N
s≥k−N
Now, for any ε > 0 we can find an N0 large enough such that
sup kνr,1 − νs,1 k ≤
r,s≥N0
1−λ
ε
2
and
∞
X
k=N0
λk <
ε
,
2kν1 k
which tells us that for h, k ≥ 2N0 we have
° Ã
°
!
°
° h
k
∞
X
X
°
°X h−j
εkν1 k
1−λ
k−j
k
°
ε+
= ε,
T1 (νj,1 ) −
T1 (νj,1 )°
λ
°≤
°
2
2kν
1k
°
°j=0
j=0
k=0
and thus γ1 is a Cauchy sequence.
The same is true for γ2 . In fact we have

kγh,2 − γk,2 k ≤ 
∞
X
j=1

λj  sup kνh+j,2 − νk+j,2 k;
j≥1
thus, since the sequence ν2 is Cauchy, so is γ2 .
Therefore g can be defined as g = π̃2 ◦ F̃ −1 |E1 (r)×{O} where π̃2 is the projection of E1 × E2 × C onto E2 .
Since we have shown above that such a g is as smooth as F̃, which in turn is as smooth as f , we are done.
Corollary 3.1.8: Let E = E1 ⊕ E2 be a splitting of a Banach space E. Then:
(i) Let f : E1 (r)×E2 (r) → E be a C 1 map fixing the origin and such that dfO is a λ-hyperbolic automorphism
of E. Assume that Lip(f − dfO ) < ε, where ε = ε(λ) is given by the previous theorem; this can be
achieved by choosing r small enough. Then the Lipschitz map g: E1 (r) → E2 (r) given by the previous
theorem satisfies g(O) = O and DgO = O. Consequently, W s (f, r) is tangent to E1 at O.
(ii) Given a λ-hyperbolic automorphism T of E, let
©
k
:= f : E1 (r) × E2 (r) → E | Lip(f − T ) < ε, kf (O)k < δ, f is C k
Nε,δ
ª
and dk f is bounded and uniformly continuous in the C k topology ,
¢
¡
k
→ C k E1 (r), E2 (r) sending f to g is
where ε and δ are given by Theorem 3.1.7. Then the map Ξ: Nε,δ
continuous.
Proof : (i) Since f (O) = O, it is obvious that O ∈ W s (f, r) and that g(O) = O. Moreover,
dgO = π̃2 ◦ d(F̃ −1 |E1 (r)×{0} )O ,
−1
= T̃ −1 , where we are setting T = dfO . So for v1 ∈ E1 we have
and d(F̃ −1 )O = dF̃O
∞
X
T2−j (O) = O.
dgO (v1 ) = π̃2 ◦ T̃ −1 (v1 , O) =
j=1
Thus the tangent space to W s (f, r) at the point O is nothing else but E1 .
(ii) It is a consequence of the following two facts, which follows easily arguing as at the end of the
previous proof:
k
then dk F̃ is uniformly continuous, bounded, and the map f 7−→ F̃ is continuous in the C k
(a) If f ∈ Nε,δ
topology.
(b) The map F̃ 7−→ F̃ −1 is continuous in the C k topology on the set of F̃ whose k-th derivative is uniformly
continuous and bounded.
52
Sistemi Dinamici Discreti, A.A. 2005/06
We are finally ready for the
Proof of Theorem 3.1.1: Let E = Tp M , and set E1 = E s (dfp ) and E2 = E u (dfp ). Since the statements (i)(v) are local and (vi) is an obvious consequence of the previous ones, we can identify a neighbourhood of p
in M with a neighbourhood of the origin in E, and replace f : M → M by a map, still denoted by the same
letter, f : E1 (r) × E2 (r) → E, with r > 0 small enough. Furthermore, shrinking r if necessary, we can assume
that Lip(f − dfp ) < ε, where ε > 0 is given by Theorem 3.1.7. Then Theorem 3.1.1 is a consequence of
Theorem 3.1.7, Proposition 3.1.3, Corollary 3.1.8 and Theorem 1.2.1.
3.2 Hyperbolic sets
In this section we collect some of the most important properties of hyperbolic sets.
Definition 3.2.1: Let M be a smooth manifold, and f : M → M a C 1 diffeomorphism. A compact completely
f -invariant set Λ ⊆ M is a hyperbolic set for the map f (and the dynamical system (Λ, f |Λ ) is a hyperbolic
dynamical system) if for any Riemannian metric on M there exist 0 < λ < 1 < µ and numbers 0 < c < C
such that for any x ∈ Λ there is a splitting Tx M = Exs ⊕ Exu so that
(i) dfx (Exs ) = Efs (x) and dfx (Exu ) = Efu(x) for every x ∈ Λ;
(ii) kdfxk (v)k ≤ cλk kvk for every x ∈ Λ, k ∈ N and v ∈ Exs ;
(iii) kdfxk (v)k ≥ Cµk kvk for every x ∈ Λ, k ∈ N and v ∈ Exu .
Definition 3.2.2: An Anosov diffeomorphism is a C 1 diffeomorphism f of a compact manifold M such that M
is a hyperbolic set for f .
Clearly a hyperbolic periodic orbit is an example of hyperbolic set. On the other hand, the hyperbolic
automorphism FL of the 2-torus discussed in Section 1.5 is an Anosov diffeomorphism: the invariant splitting
is obtained by translating to each point the eigenspaces of the matrix L.
Another example of hyperbolic set is the set Λ inside the horseshoe (see Section 2.3); in this case the
splitting is given by the vertical and horizontal directions.
Proposition 3.2.1: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M . Then the
s/u
subspaces Ex depend continuously on x and have locally constant dimensions.
Corollary 3.2.2: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M . Then the subspaces Exs and Exu are uniformly transverse, that is there is α0 > 0 such that for any x ∈ Λ, u ∈ Exs
and v ∈ Exu the angle between u and v is at least α0 .
Possibly the main feature of hyperbolic sets is the following stable manifold theorem:
Theorem 3.2.3: Let Λ ⊆ M be a hyperbolic set for a C r diffeomorphism f : M → M of a smooth manifold M , with r ≥ 1. Then there is δ > 0 such that for each x ∈ Λ we have:
(i) the local stable and unstable sets, given by W s (x, δ) = W s (x)∩B(x, δ) and W u (x, δ) = W u (x)∩B(x, δ),
s/u
r
are
intersecting
only
Tx W¢ s/u (x) = Ex ;
¡ embedded
¢ C sdisks
¢ such
¡
¢
¡ u at x and
¡ that
s
−1
u
−1
(ii) f W (x, δ) ⊆ W f (x), δ and f
W (x, δ) ⊆ W f (x), δ ;
(iii) for every ε > 0 there is Cε > 0 such that
¢
¡
d f k (x), f k (y) ≤ Cε (λ + ε)k d(x, y)
¡
¢
d f −k (x), f −k (y) ≤ Cε (µ − ε)−k d(x, y)
for all y ∈ W s (x, δ),
for all y ∈ W u (x, δ),
for all k ∈ N, where 0 < λ < 1 < µ are the costants¡in Definition¢3.2.1;
(iv) a point y ∈ M
belongs to W¢s (x, δ) if and only if d f k (x), f k (y) ≤ δ for all k ∈ N, and to W u (x, δ) if
¡ −k
and only if d f (x), f −k (y) ≤ δ for all k ∈ N;
¢
¡ s k
S
−k
(v) the global stable and
unstable ¢sets are given by W s (x) =
W (f (x), δ)
k∈N f
¡
S
and W u (x) = k∈N f k W u (f −k (x), δ) , and thus are C r immersed smooth manifolds.
We explicitely remark that, locally, stable and unstable manifolds intersect in at most one point:
3.2
Hyperbolic sets
53
Proposition 3.2.4: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M of a smooth
manifold M . Then:
(i) if δ > 0 is small enough then for any x, y ∈ Λ the intersection W s (x, δ) ∩ W u (y, δ) consists of at most
one point;
(ii) there is ε > 0 so that if x, y ∈ Λ are such that d(x, y) < ε then W s (x, δ) ∩ W u (y, δ) 6= ∅.
In particular, for δ > 0 small enough we have a map
(x, y) 7→ [x, y]δ = W s (x, δ) ∩ W u (y, δ)
defined on all pairs such that d(x, y) < δ; it can be proven that this map is continuous.
Definition 3.2.3: We say that a hyperbolic set Λ has a local product structure if there are δ, ε > 0 such
that [x, y]δ ∈ Λ for all x, y ∈ Λ with d(x, y) < ε.
Definition 3.2.4: Let (X, f ) be a dynamical
system¢on a metric space X, and ε > 0. An ε-pseudo orbit is
¡
a sequence {xk }k∈Z ⊂ X such that d f (xk ),¡xk+1 < ε for
¢ all k ∈ Z. A segment of ε-pseudo orbit is a
finite sequence {x0 , . . . , xm } ⊂ X such that d f (xk−1 ), xk < ε for all k = 0, . . . , m − 1. If {x0 , . . . , xm } is
a segment of ε-pseudo orbit such that xm = x0 , we say that {x0 , . . . , xm } is a periodic ε-pseudo orbit, and
that x0 is ε-pseudo periodic.
Theorem 3.2.5: (Anosov closing lemma) Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism
f : M → M of a smooth manifold M . Then there exist an open neighbourhood V ⊇ Λ and constants C > 1
and ε0 > 0 such that for any ε < ε0 and ¡any periodic
¢ ε-pseudo orbit {x0 , . . . , xm } ⊂ V there is a periodic
point y ∈ M of period m and such that d f k (y), xk < Cε for k = 0, . . . , m − 1.
A particular instance of periodic ε-pseudo orbit is an orbit segment x0 , f (x0 ), . . . , f m−1 (x0 ) such
that d(f m (x0 ), x0 ) < ε. For instance, such a segment exists if x0 is recurrent, and thus a consequence
of the previous Theorem is that close to any recurrent point x0 ∈ Λ there is a periodic point. Unfortunately,
in general this periodic point might not belong to Λ. A notable exception is the case of locally maximal
hyperbolic sets, that we now discuss.
Clearly, every closed completely invariant subset of a hyperbolic set is still a hyperbolic set. On the
other hand, a hyperbolic set might be a subset of a larger hyperbolic set:
Proposition 3.2.6: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M . Then there exists
an open neighbourhood V ⊃ Λ suchTthat for any C 1 diffeomorphism g: M → M sufficiently C 1 -close to f
the completely g-invariant set ΛgV = m∈Z g m (V ) is hyperbolic for g, if not empty. In particular, ΛfV ⊇ Λ is
hyperbolic.
Remark 3.2.1.
rem 3.2.17.
We have not yet proved that ΛgV is not empty. This will be a consequence of Theo-
Corollary 3.2.7: The family of Anosov diffeomorphisms of a compact manifold M is open with respect to
the C 1 topology.
Definition 3.2.5: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M . Then Λ is locally
maximal (or basic) if there is an open neighbourhood V of Λ such that ΛfV = Λ.
For instance, the discussion in Section 2.3 implies that the hyperbolic set in the horseshoe is locally
maximal; and it is not difficult to prove that a hyperbolic fixed point (and thus a hyperbolic periodic orbit)
is locally maximal.
Theorem 3.2.8: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M of a smooth manifold M . Then Λ is locally maximal if and only if it has a local product structure.
Proposition 3.2.9: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M
of a smooth manifold M . Then the periodic points are dense in N W (f |Λ ).
Corollary 3.2.10: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M
of a smooth manifold M , and assume that f is topologically transitive on N W (f |Λ ). Then f is chaotic
on N W (f |Λ ).
Definition 3.2.6: A C 1 -diffeomorphism f : M → M is Axiom A if N W (f ) is hyperbolic and periodic points
are dense in it.
54
Sistemi Dinamici Discreti, A.A. 2005/06
Remark 3.2.2. In general it is possible, even for locally maximal hyperbolic sets, to have N W (f |Λ ) 6= Λ;
see the Exercise 3.2.1.
Remark 3.2.3.
It is not known whether any hyperbolic set can be imbedded into a locally maximal one.
Exercise 3.2.1. Let f : X̃ → X̃ be the horseshoe map defined in Section 2.3, and let Λ = N W (f ) \ {p0 }.
Let ΛN
0 be the subset of Λ given by the points whose coding contains at least N 0’s between any pair of 1’s,
and let Λ0 be the subset of points whose coding contains at most a single 1. Prove that every ΛN
0 is a locally
maximal hyperbolic set for f , and that Λ0 is a hyperbolic set for f which is not locally maximal.
Exercise 3.2.2. Let f : X̃ → X̃ be the horseshoe map defined in Section 2.3, and let Λ = N W (f ) \ {p0 }.
Find a locally maximal hyperbolic subset of Λ where periodic points are not dense.
In general, f is not topologically transitive on N W (f |Λ ), but we can decompose the nonwandering set
in a finite disjoint union of completely invariant closed subsets where f is topologically transitive — and
thus chaotic.
Lemma 3.2.11: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M of
a smooth manifold M . Then N W (f |Λ ) is closed under the local product structure of Λ. In particular,
N W (f |Λ ) is a locally maximal hyperbolic set for f .
And then:
Theorem 3.2.12: (Spectral decomposition theorem) Let Λ ⊆ M be a locally maximal hyperbolic set for a
C 1 diffeomorphism f : M → M of a manifold M . Then there is a decomposition N W (f |Λ ) = P1 ∪ · · · ∪ Pr
into disjoint closed sets such that:
(i) every Pj is a completely f -invariant, locally maximal hyperbolic set for f , and f |Pj is chaotic;
(ii) for every j = 1, . . . , r there is a decomposition Pj = X1,j ∪ · · · ∪ Xsj ,j into disjoint closed subsets such
that f (Xi,j ) = Xi+1,j for i = 1, . . . , sj − 1 and f (Xsj ,j ) = X1,j ;
(iii) every Xi,j is locally maximal for f sj , and f sj is topologically mixing on it;
(iv) the sets Pj and Xi,j are unique up to indexing.
Furthermore, every Xi,j is of the form W u (p) ∩ N W (f |Λ ) for a suitable periodic point p ∈ N W (f |Λ ).
Corollary 3.2.13: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M of
a smooth manifold M . Assume that f |Λ is topologically mixing; then periodic points are dense in Λ, and
the unstable manifold of every periodic point is dense in Λ.
Corollary 3.2.14: Let f : M → M be an Anosov diffeomorphism of a connected smooth manifold M . Then
f is chaotic on M if and only if N W (f ) = M .
Remark 3.2.4. It is not known whether N W (f ) = M for any Anosov diffeomorphism, though it is
conjectured to be true.
The Anosov closing lemma shows that every periodic ε-pseudo orbit is closely followed by an actual
periodic orbit. A striking feature of hyperbolic sets is that this is true even for non-periodic orbits.
Definition 3.2.7: Let (X, f ) be a dynamical
system
on a metric space X. An ε-pseudo orbit {xk }k∈Z is
¢
¡
δ-shadowed by the orbit of x ∈ X if d xk , f k (x) < δ for all k ∈ Z.
Theorem 3.2.15: (Shadowing Lemma) Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M
of a smooth manifold M . Then there is a neighbourhood V of Λ such that for every δ > 0 there is ε > 0
so that every ε-pseudo orbit in V is δ-shadowed by an orbit of f . Furthermore, there is δ0 > 0 such that
if δ < δ0 then the orbit of f shadowing the given pseudo orbit is unique.
The idea behind shadowing is that the orbits of a perturbation of a dynamical system are ε-pseudo
orbits of the original system; since they are shadowed by actual orbits, this might give a way to conjugate
the perturbated and the original systems. But for this to work we need some kind of continuous dependence
of the shadowing orbits on the ε-pseudo orbit. This is the rationale behind the
3.2
Hyperbolic sets
55
Theorem 3.2.16: (Shadowing Theorem) Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M
of a smooth manifold M . Then there exist a neighbourhood V of Λ and a neighbourhood W of f in C 1 (M, M )
such that for all δ > 0 there is ε > 0 so that: for all topological spaces X, homeomorphisms g: X → X and
continuous maps h0 : X → V , if f˜ ∈ W is a C 1 diffeomorphism so that d0 (h0 ◦ g, f˜ ◦ h0 ) < ε, then there is
a continuous h1 : X → V such that h1 ◦ g = f˜ ◦ h1 and d0 (h0 , h1 ) < δ. Furthermore, h1 is locally unique,
in the sense that there is δ0 > 0 (depending only on Λ and f ) such that if h0 : X → V is a continuous map
satisfying h0 ◦ g = f˜ ◦ h0 and d0 (h1 , h0 ) < δ0 then h0 = h1 . Finally, h1 depends continuously on f˜.
To better understand this statement, let us first show how to prove Theorem 3.2.15 using Theorem 3.2.16.
Proof of Theorem 3.2.15: Take X = Z, endowed with the discrete topology; g: X → X given by
h1 : X → V such
g(k) = k + 1; h0 : X → V given by h0 (k) = xk ; and f˜ = f . Then Theorem 3.2.16
¡ yields
¢
means
h
(k
+
1)
=
f
h
(k)
for
all k — that is
that h1 ◦ g = f ◦ h1 and d0 (h0 , h1 ) < δ, ¡that translated
1
1
¢
k
k
h1 (k) = f (x), where x = h1 (0) —, and d xk , f (x) < δ for all k ∈ Z, as requested.
It is interesting to notice that the Anosov closing lemma is a consequence of Theorem 3.2.16 too: it
suffices to choose X = Zm , g(k) = k + 1 (mod m), h0 (k) = xk and f˜ = f .
The Shadowing Theorem is a wonderful tool for proving structural stability. In particular, we are able
to prove the strong structural stability of hyperbolic sets:
Theorem 3.2.17: Let Λ ⊆ M be a hyperbolic set for a C 1 diffeomorphism f : M → M of a smooth
manifold M . Then for every open neighbourhood V of Λ and every η > 0 there exists a neighbourhood W
of f in C 1 (M, M ) such that for all diffeomorphisms f˜ ∈ W there is a hyperbolic set Λ̃ ⊂ V for f˜ and a
homeomorphism h: Λ → Λ̃ with h ◦ f |Λ = f˜|Λ̃ ◦ h and d0 (id, h) + d0 (id, h−1 ) < η. Furthermore, h is unique
if δ is small enough.
Proof : The proof consists in three applications of the Shadowing Theorem 3.2.16. First of all, we apply
Theorem 3.2.16 taking δ < min{δ0 /2, η/2}, X = Λ, h0 = idΛ and g = f to get a neighbourhood V1 ⊂ V
of Λ, a neighbourhood W1 of f (such that d0 (f˜, f ) < ε for all f˜ ∈ W1 ) and a unique h1 : Λ → V1 such that
h1 ◦ f = f˜ ◦ h1 and d0 (idΛ , h1 ) < δ. In particular, Λ̃ = h1 (Λ) is completely f˜-invariant and hyperbolic (up
to shrinking W1 if necessary) by Proposition 3.2.6.
To prove that h1 is injective, we apply Theorem 3.2.16 taking δ as before, X = Λ̃, h0 = idΛ̃ and g = f˜;
we explicitely remark that from the proof of Theorem 3.2.16 we infer that we get the same neighbourhood W1
as soon as ε is small enough. Then we have a unique h2 : Λ̃ → V such that h2 ◦ f˜ = f ◦ h2 and d0 (idΛ̃ , h2 ) < δ.
To end the proof it suffices to show that h2 ◦ h1 = idΛ . We apply again Theorem 3.2.16 with X = Λ,
h0 = idΛ̃ and g = f˜ = f . Since
d0 (idΛ , h2 ◦ h1 ) ≤ d0 (idΛ , h1 ) + d0 (h1 , h2 ◦ h1 ) = d0 (idΛ , h1 ) + d0 (idΛ̃ , h2 ) < 2δ < δ0 ,
we can apply the uniqueness statement in Theorem 3.2.16 to get h2 ◦ h1 = idΛ , because they both commute
with f and are close to h1 .
Corollary 3.2.18: Anosov diffeomorphisms of a compact manifold M are strongly structurally stable.
Proof : The idea is to apply the previous proof with Λ = M . We get a homeomorphism h1 : M → Λ̃ ⊆ M
close to the identity and such that h1 ◦ f = f˜ ◦ h1 . We then apply the second step of the previous proof
with Λ = M again to get another map h̃2 : M → M close to the identity and such that h2 ◦ f˜ = f ◦ h2 . But
then the third step in the previous proof shows that h1 ◦ h2 = idM , and thus h1 is surjective, proving that
Λ̃ = M and that h is a homeomorphism of M , as required.
We can now give give an explicit model (up to semiconjugacy) for the dynamics on locally maximal
hyperbolic sets:
Theorem 3.2.19: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M
of a smooth manifold M . Then f |Λ is semiconjugated a topological Markov chain σA . Furthermore, f |Λ is
topologically conjugated to a topological Markov chain if and only if Λ is totally disconnected.
The Spectral Decomposition Theorem (Theorem 3.2.12) says that given a smooth manifold M and a
locally maximal hyperbolic set Λ ⊆ M for a C 1 diffeomorphism f : M → M , we can find a finite family of
disjoint sets Xi ⊆ N W (f |Λ ), i = 1, . . . , r with the following properties:
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Sistemi Dinamici Discreti, A.A. 2005/06
(i) the sets Xi are closed and open in N W (f |Λ );
(ii) for every i = 1, . . . , r there exists a ki ≥ 0 such that Xi is a completely invariant topologically mixing
locally maximal hyperbolic set for f ki .
We shall now see why the topological mixing condition on the partition of N W (f |Λ ) is crucial.
If the diffeomorphism f is topologically mixing on Λ, Corollary 3.2.13 implies that periodic points are
dense in Λ, and that the unstable manifold of every periodic point is dense in Λ. Actually, it turns out that
all unstable manifolds are dense. More precisely:
Proposition 3.2.20: Let Λ ⊆ M be a locally maximal hyperbolic set for a C 1 diffeomorphism f : M → M
mixing.
of a smooth manifold M . Assume that f |Λ is topologically
¢ Then for every α > 0 there is an N ∈ N
¡
such that for any x, y ∈ Λ and all n ≥ N we have f n W u (x, α) ∩ W s (y, α) 6= ∅. In particular, W u (x) is
dense in Λ for every x ∈ Λ.
We are now ready to define the specification property for a discrete dynamical system.
Definition 3.2.8: Let f : X → X be a bijection of a set X. A specification S = (τ, P ) S
consists of a finite
m
collection τ = {I1 , . . . , Im } of finite intervals Ij = [aj , bj ] ⊂ Z together with a map P : j=1 Ij → X such
¢
¡
that for all 1 ≤ j ≤ m and all t1 , t2 ∈ Ij we have f t2 −t1 P (t1 ) = P (t2 ). S
We say that S parametrizes
m
the collection {P |Ij | j = 1, . . . , m} of orbit segments of f . The set T (S) = j=1 Ij is the domain of the
specification, while `(S) = bm − a1 is its length. A specification S is said to be n-spaced if aj+1 − bj > n for
all j = 1, . . . , m−1; the minimum n verifying this condition is the spacing of the specification.
¢ d)
¡ Finally, if (X,
is a metric space, then we say that a specification S is ε-shadowed by a point x ∈ X if d f n (x), P (n) < ε
for all n ∈ T (S).
Definition 3.2.9: Let (X, d) be a metric space and f : X → X a homeomorphism. Then f is said to have the
specification property if for any ε > 0 there exists M = M (ε) ∈ N such that any M -spaced specification S is
ε-shadowed by a point of X, and such that for any q ≥ M + `(S) there is a q-periodic orbit ε-shadowing S.
Our aim is to show that topologically mixing locally maximal hyperbolic set have the specification
property. So this further underlines the abundance of periodic orbits in a hyperbolic set showing that,
roughly speaking, we can construct true orbits of the dynamical system from any finite collection of finite
orbit segments, with an approximation depending only on the spacing and not on the length of the segments.
Theorem 3.2.21: (Specification Theorem) Let f : M → M be a C 1 diffeomorphism of a smooth manifold M ,
and let Λ ⊆ M be a locally maximal hyperbolic set for f . Then f |Λ is topologically mixing if and only if f |Λ
has the specification property.
Proposition 3.2.22: Any homeomorphism f : X → X of a compact metric space X with the specification
property is topologically mixing.
Theorem 3.2.23: Let f : M → M be a C 1 diffeomorphism of a smooth manifold M , and let Λ ⊆ M be a
locally maximal hyperbolic set for f such that f |Λ is topologically transitive. Then there exists N ∈ N such
that for any ε > 0 there is an M = M (ε) ∈ N such that for every finite collection C of orbit segments of f
there is an M -spaced specification S parametrizing C which is ε-shadowed by a point of Λ and ε-shadowed
by a qN -periodic orbit as soon as qN ≥ M + `(S).
We end this section quoting the Grobman-Hartman linearization theorem:
Theorem 3.2.24: (Grobman-Hartman) Let f : M → M be a C 1 diffeomorphism of a smooth manifold M ,
and p ∈ M a hyperbolic fixed point of f . Then f is topologically conjugate to dfp in a neighborhood of p.